In a small saucepan, combine the butter with ¼ cup water over high heat. When the water comes to a simmer and the butter is melted, add the flour. Reduce the heat to medium and stir until the resulting paste pulls away from the sides of the pan, 1 to 2 minutes. Remove from the heat. Allow to cool for 5 minutes. Add the egg to the pan and stir vigorously until the egg is incorporated into the flour mixture. Stir in the mashed potatoes and let cool.

Pour enough oil into a medium pot so that the oil comes 3 inches up the sides. Heat the oil to 350 degrees F.

Using two soup spoons, shape the mixture into quenelles, or 2-inch footballs. (You should end up with about 35.) Roll in the panko. Deep-fry, working in batches and turning once, until crisp, brown, and heated through, 2 to 3 minutes. Drain on paper towels and season with salt.